Temperature-dependent yield stress and wall slip behaviour of thermoresponsive Pluronic F127 hydrogels

This study explores the temperature-dependent dynamic yield stress of a triblock thermoresponsive polymer, Pluronic F127, with chemical structure (PEO)100(PPO)65(PEO)100, during the sol–gel transition. The yield stress can be defined as static, dynamic, or elastic, depending on the experimental protocol. We examine the dynamic yield stress estimation for this study, which usually entails utilizing non-Newtonian models like the Herschel–Bulkley (HB) or Bingham models to extrapolate the flow curve (shear rate against shear stress). Initially, we determine the yield stress using the HB model. However, apparent wall slip makes it difficult to calculate yield stress using conventional methods, which could lead to underestimates. To validate the existence of apparent wall slip in our trials, we carry out meticulous experiments in a range of rheometric geometries. To determine the true yield stress corrected for slip, we first use the traditional Mooney method, which requires labor-intensive steps and large sample sizes over various gaps in the parallel plate (PP) design. To overcome these drawbacks, we use a different strategy. We modify the Windhab model equation by adding slip boundary conditions to the HB equation, which allowed us to calculate the slip yield stress in addition to the true yield stress. In contrast to other typical thermoresponsive polymers like poly(N-isopropyl acrylamide) (PNIPAM), our findings demonstrate that PF127's yield stress obeys the Boltzmann equation and increases with temperature.


Introduction
Hydrogels have become a highly adaptable material with a wide range of uses in many different domains, such as biomedical engineering, tissue engineering, and drug delivery. 1Because of their exceptional capacity to experience sol-gel transitions in response to temperature uctuations, thermoresponsive hydrogels have been used in biomedical applications for over two to three decades. 2,3Poloxamers are the second most commonly used thermoresponsive polymer aer poly(N-isopropyl acrylamide) (PNIPAM).Poloxamers are typically nonionic tri-block copolymers in the form of PEO-PPO-PEO based on a hydrophilic block of poly(ethylene oxide) (PEO) and hydrophobic block of poly(propylene oxide) (PPO). 3Commercially, poloxamers are also known as Pluronics.Based on molecular weight and physical state of Pluronic, it is available in different forms such as L31, P104, P85, F127, etc., where the rst letters L, P, and F represent the physical form commercially available at room temperature as liquid, paste and aked solid, respectively and numbers represents the PPO molecular mass and PEO content. 4F127 has been used for various biomedical applications because its sol-to-gel transition temperature is close to body temperature, and its rheological properties play an essential role in different biomedical and drug delivery applications. 5ydrogels are intricate materials that exhibit neither simple liquid nor perfectly elastic solid behavior.They exhibit viscoelastic behavior with both elastic and viscous components and have mechanical behavior that differs from that of solids and liquids.As the lowest stress necessary for the material to ow, the yield stress is a critical parameter in describing the mechanical behavior of hydrogels.][8] Different methods have been proposed to determine the yield stress.The yield stress can be dened as static, dynamic, or elastic, depending on the experimental protocol. 9,10The static and elastic yield stress can be found directly from creep and oscillatory shear experiments, respectively, while dynamic yield stress can be calculated by extrapolating the ow curve (shear rate versus shear stress) by tting with non-Newtonian models such as Bingham or HB model, etc. Minimum stress at which, if one can wait sufficient time, the sample reaches a nal steady Here, we aim to investigate the temperature-dependent dynamic yield stress of PF127 during the sol-gel transition.Surprisingly, the traditional method of calculating dynamic yield stress by tting the HB model for the whole range of data could not help us with different temperatures.Following a comprehensive review of the literature, we found that apparent wall slip can underestimate true yield stress values and is one of the difficulties in computing the dynamic yield stress. 11Therefore, accurately determining the yield stress of hydrogels is crucial for optimizing their performance and understanding their rheological behavior. 12The following describes the origin and history of the apparent wall slip.
The rheological characterization of complex materials, such as colloidal gels or so microgels and others, changes when bounded on smooth surfaces compared to rough or serrated surfaces due to wall effects. 13The effects of walls in complex uids are oen interpreted as apparent wall slip.Wall slip can be well understood from a typical example when a solid block is kept between two parallel plates with smooth polished surfaces, one of which is in motion, and it is due to inadequate friction present on the smooth surface. 14Similar situations can also arise in various complex uids when they exhibit solid-like behavior under external conditions like time, temperature, shear rate, etc.For example, some complex materials behave like liquids at a low shear rate and exhibit solid-like behavior at a higher shear rate. 14In some cases, solid-like behavior is observed aer a specic time, keeping the temperature and other parameters constant. 10[15][16][17][18][19][20][21] The layer may consist of pure solvent or decient particle concentrations compared to the bulk dispersion.The literature shows a high-velocity gradient near the wall due to an apparent slip.The velocity prole in between the rheometric plate gap can be directly measured by several advanced techniques combined with conventional rheometers such as particle image velocimetry (PIV), 22 nuclear magnetic resonance (NMR) velocimetry, 23 ultrasonic speckle velocimetry (USV) 24 and dynamic light scattering (DLS), etc. 25 One can also indirectly observe the wall slip from ow curves (shear rate versus shear stress). 13Over the years, a large number of studies have investigated wall slip in complex uids in the literature.While it is difficult to include them all in Table 1, we have documented a few of them.][27][28][29][37][38][39][40] 47 This paper has not discussed the temperature-dependent behavior of thermoresponsive materials.However, Aral and Kalyon studied the time and temperature-dependent behavior of poly(butadiene-acrylonitrile-acrylic) acid terpolymers and found that the temperature-dependent yield stress decreases with increasing temperatures. 36Jalaal et al. performed a systematic study to understand the rheological phase behavior of PF127 by analyzing the ow curves at different temperatures and found that the yield stress increases as an increasing function of temperature. 49They used sandblasted parallel plates to avoid the wall-slip effect and calculated the yield stress by tting with the Herschel-Bulkley (HB) model. 49But they calculated the yield stress for PF127 for different concentrations at 5 °C intervals, and from their experiments, it was shown that for thermoresponsive hydrogels like PF127, with short intervals such as 1 °C is crucial.Therefore, we investigate the temperaturedependent yield stress of PF127 gels with 1 °C during the solgel transition.Careful observation conrms the presence of apparent slip during the experiments.
Traditionally, there are three approaches to deal with the apparent slip while calculating the yield stress; rst, the slip can be avoided by using surface modication or serrated plate or alternative geometries like vane-in-cup and helix to avoid the slip.Unfortunately, we could not avoid the slip from experiments due to experimental limitations.The second approach to calculate the true yield stress is by using Mooney's calculation.We have extensively calculated the true yield stress following Mooney's method.The third approach is to use a suitable model or theory that can include the slip and calculate the true yield stress.In this article, we modied the HB model by including the slip boundary condition and were able to calculate the true yield stress at different temperatures.Finally, we plot the dynamic yield stress as a function of temperature that was obtained using three different methods (the HB model, Mooney's plot, and our approach) and found that the yield stress of PF127 increases as a function of temperature and follows Boltzmann's equation for sigmoid curves.We also compare the yield stress of PF127 with other thermoresponsive materials such as poly(N-isopropyl acrylamide) (PNIPAM) and compare the results with PF127.

Methods
Rheology.Rheological measurements of PF127 samples were performed on an MCR 302 rheometer (Anton Paar) with stainless steel cone-plate geometry (CP) and parallel plate geometry (PP) with different diameters (CP-25, CP-60, PP-40, PP-50) in rotational and oscillatory modes.The oscillatory frequency sweep experiments were performed in the linear viscoelastic region (LVR) at constant shear stain amplitude g = 0.1%, with angular frequency (u) range from 0.1 to 100 (rad s −1 ) for different temperatures (20-26 °C) during the sol-gel transition to study the linear viscoelastic properties of PF127.The ow curve is plotted for rotational tests for different temperatures varying from 10 to 40 °C.The ow curves were measured by ramp-up in logarithmic scale in two input shear rates, one from 0.1 to 100 s −1 and another from 0.01 to 1000 s −1 .The measuring point duration is kept in variable logarithmic scale with initial time, t = 100 s, and nal t = 5 s.This approach aims to prolong the measuring points at lower shear rates while shortening them at higher rates, ensuring that transient effects of the sample are minimized or fully decayed by the end of each prolonged measuring point within the low-shear range.All experiments were carried out for PP40 with a plate gap size of 1 mm except otherwise specically not mentioned.For calculating the true yield stress following Mooney's method, we also performed measurements at different gap sizes varying from (0.1, 0.25, 0.5, and 1 mm).To conrm the wall slip phenomena in our system, we also analyzed the steady-state ow curves at different geometries (CP-25, CP-60, PP-40, PP-50) at different temperatures.

Linear viscoelasticity of PF127 at different temperatures
The linear viscoelastic properties of 20% PF127 gels were investigated by frequency sweep experiments at a minimal shear amplitude strain, i.e., at g = 0.1%.The frequency sweep data for 20% PF127 at different temperatures is plotted in Fig. 1.For 20 and 21 °C, loss modulus (G 00 ) is dominant over storage modulus (G 0 ) and for almost all frequency ranges.This represents the liquid-like (viscous) behavior of PF127.At low temperatures, the PF127 solution exhibits liquid-like behavior, and at 22 °C and above, the storage modulus (G 0 ) is dominant over the loss modulus and indicates the viscoelastic uids.At 22 °C, storage modulus (G 0 ) shows a plateau at G 0 = 200.2± 20.2 Pa, known as an elastic plateau in linear rheology.As the temperature increases, the elastic plateau values increase, and the sample behaves as viscoelastic solids, i.e., gels.We found that these samples exhibit yield stress for the steady shear rheology (ow curves), and we calculated the dynamic yield stress in the following section.

Determination of dynamic yield stress from steady-state ow curves
The ow curves, i.e., shear stress (s) and viscosity (h) as a function of shear rate ( _ g) of 20% PF127 at temperatures 23 °C, 24 °C, and 25 °C are shown in Fig. 2. As shown in Fig. 2(a), For 23 °C, shear stress (s) increases linearly as a function of shear rate ( _ g), which is typical Newtonian behavior, and this is also reected in Fig. 2(b); the viscosity is independent of the shear rate ( _ g).The samples exhibit yield stress for 24 °C and 25 °C, which can be calculated using a non-Newtonian model.We dened this temperature as the sol-gel transition temperature using the same convention as Jalaal et al. 49 (here, we have used a 20% PF127 sample, and other concentrations of PF127 are shown in ESI Fig. S1 and S2 †).
For this work to calculate the yield stress, we have used the well-known Herschel-Bulkley (HB) model.The HB model with the constituent equation is given as follows, where s is the shear stress, _ g is the shear rate, s y is the yield stress, k is the consistency index, and n is the ow index.The HB model is tted to the ow curves at different temperatures, and from Fig. 3, we observed that the HB model explains the data only for 24 °C and 25 °C (Fig. 3(a) and (b)).For 26 °C and 27 °C, it fails to explain the data (the yield stress (y 0 ) value in the tting parameter table shown in the gure index shows 0 ± 49.5 and 0 ± 32.9, for 26 °C and 27 °C, respectively, which has no meaning), and, we observe a slope change in the mid shear range for both data.Therefore, to understand these results, we took help from literature, and in this context, we found one exciting work by Georgios C. Georgiou. 46In his paper, they generated the ow curves numerically for parallel plate conguration by considering various aspects of wall slip.
They discussed that the following slip law must be employed to calculate true yield stress when a wall slip occurs during the measurements.
where, v w is the relative velocity of the uids with respect to the wall, s w is the wall shear stress, s s is the slip yield stress, b is the slip coefficient, and s is the slip exponent.Fig. S3 † summarizes their works.Fig. S3(a and c   and d).Therefore, inspired by the above discussion, we realized that the HB model fails to explain the data due to wall slip.Therefore, we performed more careful experiments to conrm the presence of wall slip in our system, discussed in the following section.

Conrmation of the presence of wall slip in the system
To verify whether apparent wall slip exists in our system, we have conducted additional tests using various rheometer geometries, including parallel plate (PP) and cone and plate (CP).The literature claims that rheological measurements do not depend on measuring geometry; however, these values would vary if a wall slip is present. 13,50As seen in Fig. 4, we observed a signicant change in the ow curves from our experiments with various geometries, conrming that the sample experiences wall slip during rheological measurement.One possible reason is that in cone and plate geometry, the slip tends to occur primarily at the outer edge of the cone, where the sample comes in contact with the the stationary plate. 51This can lead to distortion in the ow prole, affecting the measured rheological properties such as viscosity and shear stress.In parallel plate geometry, wall slip typically occurs at both surfaces.Depending on the severity of the slip, it can signicantly inuence the ow behavior, especially at lower shear rates where slip effects are more pronounced. 51s discussed, wall slip usually occurs in solids due to inadequate friction on the smooth surface.Aer 24 °C, a phase transition occurs from sol-gel for 20% PF127, and at higher temperatures, the sample shows solid-like behavior, as conrmed by Fig. 4. Therefore, the signature of wall slip is seen at higher temperatures.As a result, the sample deviates from the HB Model.Using a rough surface to apply more friction and ignore the slip is the proper way to deal with it.However, there was no other geometry that we could have used to prevent slip because of experimental constraints.Consequently, we have used Mooney's traditional method to calculate the true yield stress corrected from slip using different measurements at different parallel plate (PP) geometry gaps.

Measurement of true yield stress corrected from slip by Mooney's plot using parallel plates
According to the literature, in the presence of wall slip, the shear rate measured by the rheometer differs from the true shear rate and is dened as the apparent shear rate. 23,30,31,47,52 where, _ g app is the apparent shear rate, _ g is the true shear rate corrected from slip and V S is the apparent wall-slip velocity, and h is the separation gap between the parallel plates.The true yield stress (s true ) can be calculated from the plot between (1/h) and _ g app from eqn (3).A straight line gives the slope (2V S ) and intercept ( _ g).This is typically known as Mooney's plot.As discussed in previous literature, we have rst performed steadystate ow curves at different gap sizes varying from (0.1, 0.25, 0.5, and 1 mm).Then, the shear stress value and corresponding shear rate values were taken from each plot and average values (the error bar represents the mean and standard deviation of all 3 measurements in each gap size).As seen in Fig. 5 and S4, † we have plotted these curves for various temperatures and shear stresses.The tted line has a slope of 2V S ; the slip velocity and shear stress value are represented by half of the slope, and for any intercept ( _ g∼0), the corresponding shear stress is known as dynamic yield stress. 23,30,31,47,52 Though this method has traditionally been used to calculate the true yield stress of a material, eliminating the slip during the measurements, this method is labor-intensive, and a large sample is required for repetitive experiments for different gaps of rheometric geometry.Therefore, to overcome these drawbacks, we followed a second approach, where we included the slip boundary conditions in the HB equation, modied the HB equation in the form of the famous Windhab model equation, and calculated the true yield stress and slip yield stress.

Calculation of true yield stress by including slip boundary condition in HB model
Consider a simple plane coquette shearing ow between two parallel plates with a gap (h), as shown in Fig. 6(a).The lower plate is xed, and the upper plate moves at a constant velocity (V) along the x-direction.If the density of the uid is (r) and the uid velocity vector is given as u = u(u 1 , u 2 , u 3 ) where bold letter denote the vector representation and u 1 , u 2 , u 3 are the x, y, and z components of the uid velocity vector (u), then the equation of continuity (conservation of mass) 14,53 is given as Now, for steady-state conditions and incompressible uid (r is constant), the above equation will reduce to Similarly, the equation of motion (conservation of momentum), famously known as the Navier-stokes equation, 53 is given as where r is the uid density, u is the uid velocity, t is time, p is the pressure, m is the coefficients of viscosity, and F represents body forces per unit volume (i.e., external forces such as gravity or electromagnetic forces).Now, since we consider the ow is in between two plates, the velocity of the uids in one direction, i.e., along the x-direction, and no motion in the y and z-direction.Then, the velocity of the uids can be represented as u = u(u 1 , 0, 0), and since we consider a simple plane coquette shearing ow, there is no pressure gradient, i.e., Vp = 0, and we also assume that there exist no external forces (F = 0) using these conditions eqn ( 5) and ( 6) can be reduced to following, Eqn (5) can be written as Similarly, eqn (6) reduced to only the x-component momentum equation, i.e.
For simplicity, we drop the subscript and write u instead of u 1 0u(y Eqn (7) represents the velocity of the uids between parallel plates.If we apply a no-slip boundary condition (velocity of the uids and plate are same), i.e., for a lower plate which is xed u(0) = 0 and the upper plate moving with velocity (v) so, u(h) = v.Applying this to the boundary condition in eqn (7), we arrived at, This gives a linear velocity prole for no-slip boundary conditions and is shown in Fig. 6(a), if the shear rate ( _ g) is linear, then this can be dened as If we assume a monotonic slip law (i.e., slip velocity is the same for both plates), the boundary conditions, i.e., u(0) = V S and u(0 Applying the slip boundary condition eqn (7) has the following form.
The velocity prole is still linear but slightly shied, as shown in Fig. 6(b), and now, the true shear rate can be dened as Experimentally, the shear rate is measured as the apparent shear rate, where, U is the rotational speed of the rheometric geometry, R is the radius of the rheometric plate, and h is the gap between the parallel plates.Now, using eqn (11) in HB model The modied HB model, including the slip boundary condition, has the following form: app m ðmultiply and divide byðg *ÞÞ This equation is well known in the form of the Windhab model with additional power terms m and n (when m = n = 1, eqn ( 15) is reduced to the Windhab equation).Therefore, we dened this as a modied Windhab equation.To avoid confusion, we used _ g app as only _ g and if we write eqn (15) with some redened constant inspired by the Windhab model, then the equation becomes where, k = h N and k 2 = (s 1 − s 0 ) and eqn ( 16) very well ts our data.But from eqn (16), we found two yield stresses (s 0 , s 1 ) and also two exponents (n and m) and _ g* is a normalization constant.We have taken some arguments from different literature to understand each term's physical meaning and generalization of the approach to any system.
Consequently, to validate our approach generalized to any system, we took help from arguments discussed in Moud et al., 47 where they discussed wall slip in kaolin clays, as shown in Fig. 7(a) and discussed different regimes and their physical signicance.Then, we tted eqn (16) to the data from Moud et al., 47 as shown in Fig. 7(a), and we found that the equation tted very well to the data.Here s 0 value lies in Regime-II, dened by Moud et al. 47 in Fig. 7(a), which is nothing but the slip yield stress and s 1 lies in Regime-III, which is the true yield stress.Therefore, from tting parameters, we found that s 0 , is the slip yield stress, and s 1 , is the true yield stress, and _ g*, is transition shear rate.When observing the ow curves, we see slope changes from low shear to higher shear rates aer some shear rate.Here, _ g* represents the corresponding shear rate and is dened as the transition shear rate, and (n and m are slip and power-law exponents, respectively).Therefore, we write the eqn (15) as where, s s is slip yield stress and s y is the true yield stress, and all other terms are the same as dened previously.In all the ow curves we discussed, the data ranges from 0.1 to 100 s −1 , and to verify that our approach is not limited to any range, we also tted data for a large range, i.e., from 0.01 to 1000 s −1 , and is shown the Fig. S5.† We compared the values with the conventional Mooney's method to verify the true yield stress estimated from our method, in Table 2.
Temperature-dependent dynamic yield stress Fig. 9(a) shows the yield stress obtained from various methods plotted as a function of temperature.We found that yield stress increases as a function of temperature, and the experimental data following a sigmoid curve can be tted by an empirical equation in the form of the Boltzmann equation 54 as follows, where, s 00 y is yield stress plateau at low temperatures and s 0 y yield stress plateau at high temperatures and T c is the critical temperature where the yield stress increases exponentially.The yield stress values gradually increase as shown by the plateau, in this case.The increasing value of yield stress for PF127 is consistent with Jalaal et al.'s study 49 at different temperatures at 5 °C intervals but doesn't formally report any trend, and as discussed earlier near the sol-gel transition temperature, each 1 °C interval is important; therefore we carefully calculated the yield stress and found the trend.The increasing trend can be understood by the mechanism suggested by Suman et al. that the individual micelle size may increase and can form a glassy state at higher temperatures can account for the exponential increase in the yield stress at higher temperatures. 55In the current study, we compared the results with other thermoresponsive polymer poly(N-isopropyl acrylamide) (PNI-PAM) and found some interesting and contradictory results.
When comparing the yield stress for other conventional thermoresponsive polymers such as PNIPAM, whereas the temperature increased, a noticeable decrease in yield stress was shown in Fig. 9(b) (data are taken from Divoux et al. 43 and plotted).This phenomenon can be attributed to the coil-toglobule transition experienced by PNIPAM, where hydrophobic interactions cause the polymer chains to collapse from extended coils to compact globules.7][58] In contrast, Pluronic PF127 exhibited increased yield stress with increasing temperature.Above its critical micelle temperature T CMT (around 15-30 °C), Pluronic PF127 undergoes micelle formation, where hydrophobic PPO blocks assemble into micellar cores surrounded by hydrophilic PEO shells.The increase in temperature leads to enhanced micelle aggregation and packing, resulting in the formation of a denser gel network.This densication of the gel network corresponds to an increase in yield stress as the gel becomes more resistant to deformation.The contrasting temperature-dependent yield stress behaviors of PNIPAM and Pluronic PF127 highlight their distinct gelation mechanisms.While PNIPAM's coil-to-globule transition decreases yield stress, Pluronic PF127's micelle formation leads to an increase in yield stress with increasing temperature. 58The biomedical eld has used body temperature as the transition temperature for two to three decades.Examining the yield stress concerning temperature can provide an additional understanding of the rheological characteristics of PF127 in diverse drug delivery applications and transport phenomena.

Conclusion
This study thoroughly investigated the yield stress behavior of PF127 at various temperatures using the Herschel-Bulkley Further tests proved the presence of wall slip using ow curves depending on rheometric geometry, which led to the adoption of the modied Windhab model to address the limitations of the HB model.The ow curves at higher temperatures were more accurately tted (with higher R-square values) by this wall slip-accounting model.The temperature-dependent yield stress increased signicantly as the results showed, following a sigmoid curve that could be explained by an empirical equation like the Boltzmann equation.Comparative analysis with other thermoresponsive polymers, such as PNIPAM, highlighted the distinct gelation mechanisms and temperaturedependent yield stress behaviors of PF127.Unlike PNIPAM, which shows a decrease in yield stress with increasing temperature due to the coil-to-globule transition, PF127 exhibited an increase in yield stress attributed to enhanced micelle aggregation and network densication at higher temperatures.
This research emphasizes the importance of considering wall slip effects in rheological measurements and provides  a robust framework for understanding the temperaturedependent rheological properties of PF127.The ndings have signicant implications for using PF127 in biomedical applications, particularly in drug delivery systems where temperature-sensitive gelation is crucial.Future research should explore applying these models to other complex uids and further investigate the mechanisms underlying the observed temperature-dependent behaviors.
) † represents ow curves shear stress (s) vs. shear rate ( _ g) generated numerically for HB uids in case of the yield stress (s y ) = 2 Pa, slip exponent (s) = 1, and slip-coefficient (b) = 1 × 10 4 Pa s s /m s , ow index (n) = 1, and consistency index (k) = 8 × 10 −3 Pa s n for different the slip yield stress (s s ) as: (a) s s = 0; (c) s s = 1 Pa, respectively.Finite slip yield stress shown in Fig. S3(c) † is very similar to 26 °C and 27 °C data for our data shown in Fig. 3(c and d).The above arguments are also valid for investigation of the inuence of slip exponent (s) variation on the ow curve with s s = 0.5 Pa, s y = 2 Pa, b = 1 × 10 4 Pa s s /m s , n = 1 and k = 8 × 10 3 Pa s n : (b) s = 1; (d) s = 2.One can observe that for s = 1 (Fig. S3(b) †), the ow curves show a similar trend as of Fig. 3(c

Fig. 3
Fig. 3 (a-d) represents HB model fitting of flow curves for 20% PF127 at temperatures varying from 23 to 27 °C, respectively.

Fig. 5 (
a) and (b) represent Mooney's plot for 25 °C and 28 °C for different shear stress values.We observed that from Fig. 5(a) and (b) shear stress values (140.9 ± 2.1) Pa and (213.7 ± 1.6), respectively, the intercept almost approached zero; therefore, we dened these values as yield stress at that temperature.In Fig. S4, † we have shown Mooney's plot at 26-30 °C, respectively, and calculate the corresponding yield stress values.

Fig. 5
Fig. 5 (a and b) Mooney's plots for PP-40 geometry between apparent shear ( _ g app ) rate and reciprocal of the gap between two parallel plates (1/ h) for different shear stress at different temperatures from 25 to 28 °C, respectively.Different colour lines represent the straight line fit to each data set for different shear stress values, and the symbols and error bars represented here are the mean and standard deviation of three sets of measurements in each gap size.

Fig. 6
Fig. 6 Schematic representation of velocity profile between two parallel plates with a gap (h) for various boundary conditions (a) no slip (b) slip boundary condition, respectively.

Fig. 7
Fig. 7 (a) Flow curves of Kaolin clay in a parallel plate geometry with different gaps (adapted and reproduced with permission from ref. 47) and (b) data taken from (a) and plotted with eqn (16).The red and pink lines are the extrapolation of yield stress values found from fitting the vertical dotted line represented by the transition shear rate obtained from the fitting eqn (16).
model and the modied Windhab model, shedding light on the rheological characteristics of the thermoresponsive polymer.The HB model effectively described the data at lower temperatures (23 °C to 25 °C) but failed at higher temperatures (26 °C and 27 °C) due to wall slip.This failure was indicated by a slope change in the mid-shear range with a kink around a shear rate of 10 s −1 , and unrealistic yield stress values at these temperatures.

Fig. 9
Fig.9(a) Comparison of dynamic yield stress values obtained from different methods for 20% PF127.The data fits Boltzmann's equation (eqn(18)).(b) The temperature-dependent yield stress of reported data for 20% PF127 taken from Jalaal et al.49 is fitted with eqn (18), and PNIPAM gels are from ref.43.

Table 1
List of articles published on wall slip behaviour and their significance in different systemsPublication © 2024 The Author(s).Published by the Royal Society of Chemistry RSC Adv., 2024, 14, 23772-23784 | 23773

Table 1 (
Contd.)However, we were inspired by recent work by Moud et al., which evaluated apparent slip in the colloidal suspensions of kaolinite clay and proposed a generalized slip model that can be applied to many other colloidal systems.